## Tricky Equation With Factorisation

Some factorisations are not obvious. Consider the equation
$x-y^2= xy^2-x^2$

The right hand side factorises to give
$x-y^2=x(y^2-x)$

Subtract
$x(y^2-x)$
from both sides to give
$x-y^2+x(x-y^2)=0$

Now we can factorise both sides completely.
$(1+x)(x-y^2)=0$

Hence
$1+x=0 \rightarrow x=-1$
and
$y$
is arbitrary or
$x-y^2=0 \rightarrow y = \pm \sqrt{x}$
which is the graph of
$y= \sqrt{x}$
together with it's reflection in the x - axis.